By baseball hat, do you only mean baseball team hats or can it be baseball-style caps? I have one blue hat, mesh with plastic adjustment snaps, that is from a rally I went to in 1984. It says "Boston Lights the Way to Victory Mondale-Ferraro." I did not pick up another political souvenir for 8 years, I think, believing my collection was jinxed.
Baseball, I have one of the cool road Seattle hats from the year they changed the logo - Green bill, dark blue skull piece. I have 3 Red Sox - "Schilling", my lucky hat from 2007 and I think 2004, blue cotton; an old wool hat with adjustable snaps that I think dates from the early Pedro era, but could be a 1980s model; and a patriotic B fitted hat. I threw my hanging sox hat into 4 mile run the last Sunday of the regular season. For the Nats, I have a blue DC logo that I like and a red curly W home hat. I have 3 Os hats, but did not pay for any (an HTS white give away with cartoon bird, a departure gift from my Boston office that is a classic cartoon bird model - fitted, so it does not fit, and a black Miller give away with the orangutanically correct bird). I also have a NYY hat, which I bought in 2003 for Halloween, and was given a Yankee Hater hat.
I'll skip the 3 Pats, 3 Yale, Bruce Springsteen, Sam Adams hats...
edit 11/16 AM - I forgot my 3d Nats hat, the Miller Lite give away with the red bill, white and blue skull, and the new logo in front.